1. P3: Solving linear
equations and linear inequalities

2. Algebraic Properties
Properties of Equality (let u, v, w, and z be real numbers,
variables, or algebraic expressions)
1. Reflexive
u = u
2. Symmetric
If u = v, then v = u
3. Transitive
If u = v, and v = w, then u = w
4. Addition
If u = v and w = z, then u + w = v + z
5. Multiplication
If u = v and w = z, then uw = vz

3. Solving Equations A linear equation in x is one that can be written
in the form
ax + b = 0
where a and b are real numbers with a = 0
A solution of an equation in x is a value of x for
which the equation is true.
So, how many solutions are there to a linear
equation in one variable???

4. Let’s practice… Solve for the unknown:

5. Let’s practice… Solve for the unknown:8 8

6. Let’s practice… Solve for the unknown and support with grapher:
Now, how do we get graphical support???

7. A linear inequality in x is one that can be written in the form
ax + b < 0, ax + b < 0, ax + b > 0, or ax + b > 0
where a and b are real numbers with a = 0
A solution of an inequality in x is a value of x
for which the inequality is true.
The set of all solutions of an inequality is the
solution set of the inequality.

8. Properties of Inequalities
Let u, v, w, and z be real numbers, variables, or algebraic
expressions, and c a real number.
1. Transitive
If u < v and v < w, then u < w
2. Addition
If u < v, then u + w < v + w
If u < v and w < z, then u + w < v + z
3. Multiplication
If u < v and c > 0, then uc < vc
If u < v and c < 0, then uc > vc
(the above properties are true for < as well – there are
similar properties for > and >)

9. Guided Practice: Solve the inequality:
When solving inequalities, don’t
forget to switch the inequality
sign whenever you multiply or
divide by a negative number!!!

10. Guided Practice: Solve the inequality, write your answer in interval
notation, and graph the solution set: –2

11. Guided Practice
Solve the double inequality, write your answer in
interval notation, and graph the solution set:
(–7, 5] –7 5

12. 8 Whiteboard Practice: 11 – , – 7
Solve the inequality, write your answer in interval
notation, and graph the solution set:
–11/7 11 8 – , – 7

13. Whiteboard Practice:
Solve the inequality:

14. Whiteboard Practice: Solve and support with grapher:12 12
Graphical Support?

15. Homework:
p odd, odd